File size: 3,734 Bytes
695bd70 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | import Mathlib
import OAI.Geometry.SmoothYau.Smoothness.SphericalBaseProfileRadius
namespace OAI
noncomputable section
open Set Filter Manifold Bundle MeasureTheory
open scoped Topology ContDiff ENNReal
open Matrix
open scoped Topology Matrix.Norms.Elementwise
namespace YauCounterexamples
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
{M : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ, E) ∞ M]
local instance chartExtDualNorm : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance
local instance chartExtDualSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance
local instance chartExtFormNorm : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
local instance chartExtFormSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance
def euclideanMetricForm : E →L[ℝ] E →L[ℝ] ℝ := innerSL ℝ
theorem exists_compact_chart_metric_extension (g : SmoothMetric E M) (p : M)
{K : Set E} (hK : IsCompact K) (hKO : K ⊆ (chartAt E p).target) :
∃ ĝ : SmoothMetric E E, ∀ᶠ y in 𝓝ˢ K, ∀ v w : E,
ĝ.inner y v w = g.inner ((chartAt E p).symm y)
(mfderiv 𝓘(ℝ, E) 𝓘(ℝ, E) (chartAt E p).symm y v)
(mfderiv 𝓘(ℝ, E) 𝓘(ℝ, E) (chartAt E p).symm y w) := by
obtain ⟨V, hV, hKV, hVO⟩ := normal_exists_closure_subset hK.isClosed (chartAt E p).open_target hKO
obtain ⟨η, hηone, hηzero, hηrange⟩ :=
exists_contMDiffMap_one_nhds_of_subset_interior (I := 𝓘(ℝ, E)) (n := ⊤) (t := V)
hK.isClosed (by simpa only [hV.interior_eq] using hKV)
have hη : ContDiff ℝ ∞ η := contMDiff_iff_contDiff.mp η.contMDiff
have hsη : tsupport η ⊆ (chartAt E p).target := by
apply (closure_mono (show Function.support η ⊆ V from ?_)).trans hVO
intro x hx
by_contra hn
exact hx (hηzero x hn)
have hηφ : ContDiff ℝ ∞ (fun y => η y • chartMetricForm g p y) := by
rw [← contMDiff_iff_contDiff]
apply contMDiff_of_tsupport
intro y hy
have hyO := hsη (tsupport_smul_subset_left η (chartMetricForm g p) hy)
rw [contMDiffAt_iff_contDiffAt]
exact hη.contDiffAt.smul (((chartMetricForm_contDiffOn g p) y hyO).contDiffAt
((chartAt E p).open_target.mem_nhds hyO))
let B : E → E →L[ℝ] E →L[ℝ] ℝ :=
fun y => η y • chartMetricForm g p y + (1-η y) • (euclideanMetricForm (E := E))
have hB : ContDiff ℝ ∞ B := hηφ.add ((contDiff_const.sub hη).smul contDiff_const)
have hsym : ∀ y v w, B y v w = B y w v := by
intro y v w
change η y * chartMetricForm g p y v w + (1-η y) * inner ℝ v w =
η y * chartMetricForm g p y w v + (1-η y) * inner ℝ w v
rw [chartMetricForm_symm, real_inner_comm]
have hpos : ∀ y v, v ≠ 0 → 0 < B y v v := by
intro y v hv
have hip : 0 < inner ℝ v v := real_inner_self_pos.mpr hv
change 0 < η y * chartMetricForm g p y v v + (1-η y) * inner ℝ v v
by_cases hy : y ∈ (chartAt E p).target
· have hp := chartMetricForm_pos g p hy hv
obtain ⟨hη0, hη1⟩ := hηrange y
by_cases hz : η y = 0
· simpa only [hz, zero_mul, sub_zero, one_mul, zero_add] using hip
· exact add_pos_of_pos_of_nonneg (mul_pos (lt_of_le_of_ne hη0 (Ne.symm hz)) hp)
(mul_nonneg (sub_nonneg.mpr hη1) hip.le)
· have hnV : y ∉ V := fun hyV => hy (hVO (subset_closure hyV))
simpa only [hηzero y hnV, zero_mul, sub_zero, one_mul, zero_add] using hip
refine ⟨flatMetricOfForm B hB hsym hpos, ?_⟩
filter_upwards [hηone] with y hy
intro v w
change B y v w = _
rw [← chartMetricForm_pairing]
simp [B, hy]
end YauCounterexamples
end
end OAI
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